# Q: What is the prime factorization of the number 123,456,788?

A:
• The prime factors are: 2 x 2 x 7 x 13 x 17 x 71 x 281
• or also written as { 2, 2, 7, 13, 17, 71, 281 }
• Written in exponential form: 22 x 71 x 131 x 171 x 711 x 2811

## Why is the prime factorization of 123,456,788 written as 22 x 71 x 131 x 171 x 711 x 2811?

### What is prime factorization?

Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

### Finding the prime factors of 123,456,788

To find the prime factors, you start by dividing the number by the first prime number, which is 2. If there is not a remainder, meaning you can divide evenly, then 2 is a factor of the number. Continue dividing by 2 until you cannot divide evenly anymore. Write down how many 2's you were able to divide by evenly. Now try dividing by the next prime factor, which is 3. The goal is to get to a quotient of 1.

#### If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 123,456,788 by 2

123,456,788 ÷ 2 = 61,728,394 - No remainder! 2 is one of the factors!
61,728,394 ÷ 2 = 30,864,197 - No remainder! 2 is one of the factors!
30,864,197 ÷ 2 = 15,432,098.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
30,864,197 ÷ 3 = 10,288,065.6667 - This has a remainder. 3 is not a factor.
30,864,197 ÷ 5 = 6,172,839.4 - This has a remainder. 5 is not a factor.
30,864,197 ÷ 7 = 4,409,171 - No remainder! 7 is one of the factors!
4,409,171 ÷ 7 = 629,881.5714 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
4,409,171 ÷ 11 = 400,833.7273 - This has a remainder. 11 is not a factor.
4,409,171 ÷ 13 = 339,167 - No remainder! 13 is one of the factors!
339,167 ÷ 13 = 26,089.7692 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
339,167 ÷ 17 = 19,951 - No remainder! 17 is one of the factors!
19,951 ÷ 17 = 1,173.5882 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
19,951 ÷ 19 = 1,050.0526 - This has a remainder. 19 is not a factor.
19,951 ÷ 23 = 867.4348 - This has a remainder. 23 is not a factor.
19,951 ÷ 29 = 687.9655 - This has a remainder. 29 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
19,951 ÷ 71 = 281 - No remainder! 71 is one of the factors!
281 ÷ 71 = 3.9577 - There is a remainder. We can't divide by 71 evenly anymore. Let's try the next prime number
281 ÷ 73 = 3.8493 - This has a remainder. 73 is not a factor.
281 ÷ 79 = 3.557 - This has a remainder. 79 is not a factor.
281 ÷ 83 = 3.3855 - This has a remainder. 83 is not a factor.
...
Keep trying increasingly larger numbers until you find one that divides evenly.
...
281 ÷ 281 = 1 - No remainder! 281 is one of the factors!

The orange divisor(s) above are the prime factors of the number 123,456,788. If we put all of it together we have the factors 2 x 2 x 7 x 13 x 17 x 71 x 281 = 123,456,788. It can also be written in exponential form as 22 x 71 x 131 x 171 x 711 x 2811.

### Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 123,456,788.

 123,456,788 2 61,728,394 2 30,864,197 7 4,409,171 13 339,167 17 19,951 71 281

### More Prime Factorization Examples

 123,456,786 123,456,787 123,456,789 123,456,790 21 x 31 x 20,576,1311 312 x 128,4671 32 x 3,6071 x 3,8031 21 x 51 x 371 x 333,6671

Try the factor calculator

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